Limits and Continuity Take Home
Name:_____________________
1. Use the graph to find lim f ( x ) for
7. If lim f ( x )  
x 1
x c
1
and lim g ( x ) 
x c
2
.
8. Find the limit: lim
x2
x2
x 2
x2
3
x 8
x0
for f ( x ) 
1
x 1
x c
f ( x)
.
g( x)
.
x9 3
10. Find the limit: lim
2. Use the graph of f to find
3
, find lim
.
2
x 4
9. Find the limit: lim
2
.
x
2
.
1  cos x
11. Find the limit: lim
x 0
x
12. Find the limit: lim
x 0
.
x
.
sin 3 x
2
x  5 x  14
.
x7
a. Find lim f(x) (if it exists).
13. Let f(x) =
x 7
b. Identify another function that agrees with f(x) at all but one point.
c. Sketch the graph of f(x).
3. Use the graph to find lim f ( x ) (if it exists).
x0
2
14. Find the limit: lim
1 2x  1
x 1
x 1
1
15. Find the limit: lim x  3
x 0
x
16. Find the limit:
4. Let
3 x  1, x  1
R
||
. Find the limit:
f ( x)  S
|| 3 x , x  1
T2
2
lim f ( x ).
17. Find the limit:
x 1
18. Find the limit:
exist.
2
6. Find the limit: lim ( 2 x  6 x  1).
x2
19. Find the limit:
1
3.
lim
2 x  1.
lim
x  1.
lim
1
.
x 1
lim
7 x  56
.
8 x
x2 
x1
x1
5. Sketch the graph of a function y = f(x) such that lim f(x) does not
x 2

.
x8
Limits and Continuity Take Home
20. Let
a.
f ( x) 
x3
.
x3
Name:_____________________
32 For
X
F(x)
lim f ( x)
x3
b.
c.
complete the table then use the table to approximate
Find each limit (if it exists).
2
lim f ( x)
x3
lim f ( x)
33. Given
x3
and
34. Let
and
35. Find
.
21. Find the x-values (if any) for which f is not continuous.
22. Find the x-values (if any) at which
f ( x) 
x 2  6x  5
x6
is
discontinuous. Are they removable or non-removable?
23. Determine the value of c so that f(x) is continuous on the entire real
line if
24. Use the Intermediate Value Theorem to show that the function f(x) = x4
 2x2 + 3x has a zero in the interval [2, 1].
25. Use a graphing utility to graph f(x) = x3  2x  5. Then use the graph
to find the interval for which the Intermediate Value Theorem guarantees
the existence of at least one number c in that interval for which f(c) = 0.
26. Find all vertical asymptotes of the graph of
f ( x) 
find
.
2x  2
.
( x  1)( x 2  x  1)
27. f(x) decreases without bound as x approaches what value from the
right?
f ( x) 
7
( x  1)(7  x)
28. Find the horizontal asymptote for
29. Find the limit:
lim
x 
x2
4x2  1
2x2  x  7
f ( x) 
.
x2  1
.
30. Find the limit:
31. Find all horizontal asymptotes for f ( x ) 
4x
x2  9
.
find